Appendix B - Dimensions of units of magnetism

Summary

SI magnetic units are easily related to the current, voltage and energy in MKS units, since the SI system was originally developed under the assumption that magnetism is originated from electric current. The dimensions of magnetic units are shown below; A = amperes, s = seconds, kg = kilograms and m = meters.

1. (1) newton (N) = kg m/s²;

2. (2) joule (J) = kg m²/s²;

3. (3) magnetic field (H) = A/m;

4. (4) henry (h) = kg m²/s²A²;

5. (5) tesla (T) = kg/s²A;

6. (6) weber (Wb) = kg m²/s²A.

Magnetism in cgs units is less transparent. The unit of magnetic moment m is the emu. The density of the magnetic moment M_S is emu/cm³. The magnetic induction is given by B = H + 4πM_S, where the magnetic field H is given in units of oersted (Oe) and B is given in gauss (G). (1 Oe = 1 G in air, since M = 0 in air.) Thus, the “4π” in 4πM_S is not dimensionless. Its dimension is [G/(emu/cm³)], and it is equivalent to the inverse of susceptibility, so that the unit of 4πM_S is [G/(emu/cm³)] · [emu/cm³] = G.

The dimension of “emu” can be understood from the dimension of energy. In Chapter 2, we discussed the magnetostatic energy per unit volume of magnetic material under a magnetic field H to be ∼M_S · H. The dimension of M_S · H is [emu · cm⁻³] · [Oe], which should be equivalent to [erg · cm⁻³]. Therefore, the dimension of emu is [erg/Oe] or [emu/G] in air.